This course theoretically extends and complements the concurrently taught course KMA/PAN2. Emphasis is placed on precise mathematical statements, proofs of selected theorems, counterexamples, and connections between theoretical results and computational applications. 1 - The Riemann integral. Partitions of an interval, Riemann sums, and the definition of Riemann integrability. 2 - Fundamental properties of the Riemann integral. Linearity, monotonicity, additivity with respect to the interval of integration, and integrability criteria. Integrability of continuous and monotone functions. 3 - Mean value theorems for integrals. The integral as a function of its upper limit, the fundamental theorem of calculus, and the Newton-Leibniz formula. 4 - Antiderivatives. Integration by substitution and integration by parts. 5 - Integration of rational functions and selected expressions involving trigonometric functions and radicals. Geometrical applications: areas of plane regions, volumes of solids of revolution, and lengths of curves. 6 - Improper integrals over unbounded intervals and integrals of unbounded functions. Convergence, absolute convergence, and comparison tests. 7 - Numerical series. Sequences of partial sums, the Cauchy convergence criterion, a necessary condition for convergence, and fundamental examples. 8 - Series with non-negative terms. Comparison, ratio, root, and integral tests. 9 - Alternating series, the Leibniz test, and absolute and conditional convergence. 10 - Power series over the real numbers. Radius and interval of convergence, analysis of endpoints, and fundamental properties within the interval of convergence. 11 - Taylor polynomials of functions of one variable, Taylor's theorem with remainder, error estimates, and Taylor series of selected elementary functions. 12 - Functions of two real variables. Fundamental concepts, limits, and continuity. 13 - Partial derivatives, differentiability, the total differential, differentiation of composite functions, the gradient, tangent planes, and linear approximation for functions of two real variables. 14 - Second-order Taylor polynomials and local extrema of functions of two variables. The implicit function theorem for an equation F(x,y)=0, its geometrical interpretation, and basic applications.
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Self-study (text study, reading, problematic tasks, practical tasks, experiments, research, written assignments), Lecture, Practicum
- Class attendance
- 56 hours per semester
- Preparation for credit
- 28 hours per semester
- Preparation for exam
- 28 hours per semester
- Home preparation for classes
- 38 hours per semester
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